Hyers-Ulam Type Stability of the Pexiderized Cauchy Functional Equation in Locally Convex Cones
Jafar Mohammadpour, Abbas Najati, Iz-iddine EL-Fassi
TL;DR
This paper extends Hyers-Ulam stability theory to the Pexiderized Cauchy equation $f(x+y)=g(x)+h(y)$ within locally convex cones, using the symmetric topology and complete separated uc-cones. It proves a main existence-uniqueness result: under a bounded neighborhood condition $f(x+y)\in v(g(x)+h(y))v$, there exists a unique additive $A:\mathscr{P}_1\to\mathscr{P}_2$ with $A(x)$ close to $f,g,h$ in cone neighborhoods, and it provides explicit bounds. The work derives several corollaries aligned with NR1/NR2 stability results and culminates in a comprehensive Hyers-Ulam stability statement for cone-valued mappings, highlighting methodological extensions from Banach-space settings to ordered cone frameworks. These results enhance stability analysis in ordered vector spaces and have potential implications for functional equations on cones and related operator theory.
Abstract
The foundation of locally convex cone theory relies on order-theoretic concepts that induce specific topological frameworks. Within this structure, cones naturally possess three distinct topologies: lower, upper, and symmetric. In this paper, we consider the Hyers-Ulam type stability of the Pexiderized Cauchy functional equation $f(x+y)=g(x)+h(y)$ in locally convex cones. Additionally, we present several significant corollaries that follow from our primary findings.